Economic Consequences of Nth-Degree Risk
Increases and Nth-Degree Risk Attitudes
Elyès Jouini
Ceremade
Clotilde Napp
CNRS-DRM
Diego Nocetti
Clarkson University
May 11, 2012
AbstractWe study comparative statics of Nth-degree risk increases, as de…ned by Ekern (1980), within a large class of problems that involve bidimen-sional payo¤s and additive or multiplicative risks. We establish necessary and su¢ cient conditions for unambiguous impact of Nth-degree risk in-creases on optimal decision making. We develop a simple and intuitive approach to interpret these conditions : novel notions of directional Nth-degree risk aversion that are characterized via preferences over lotteries
1
Introduction
Consumers select how much to save, how much to invest in di¤erent assets, how long to work, and how much to spend on medical care under a great degree of uncertainty. Firms invest large amounts of money in risky endeavors. Policy-makers allocate scarce resources to projects with highly uncertain returns (e.g. environmental and health care projects). A natural question that arises in these and other similar problems is the following: How does an increase in risk a¤ect the optimal level of exposure to the risk? In this paper we revisit this question within a large class of problems that involve 1) a bivariate utility function, 2) a linear constraint that links the two attributes that enter the decision-maker’s utility, and 3) increases in Nth-degree risk as de…ned by Ekern (1980).
A substantial amount of research has been devoted to analyze closely related problems. The general setting that we use is in the spirit of Dardanoni (1988), who developed a unifying framework to analyze the impact of increases in risk à la Rothschild and Stiglitz (1970) in a large class of problems. More recently, a number of authors have analyzed the impact of a larger class of increases in risk - namely, Nth degree increases in risk as in Ekern (1980) - in di¤erent frame-works, including the classical problems of precautionary saving (Eeckhoudt and Schlesinger, 2008), precautionary labor supply (Chiu and Eeckhoudt, 2010), and portfolio choice (Chiu et al., 2011)1. The focus of the recent work and much of
the earlier literature has been on …nding su¢ cient conditions for unambiguous comparative statics of changes in risk. Although this is an important endeavor, many important questions remain unanswered. Are these conditions also nec-essary? If not, what are the necessary and su¢ cient conditions? What is the interpretation of these conditions, which frequently involve establishing the sign of two or more third, fourth, or higher-order partial derivatives of the payo¤ function? How are these conditions related to more primitive attitudes towards risk? Our objective is to answer these questions.
We begin the analysis in the next Section by revisiting Ekern’s (1980) re-sults. Following the methods of Rothschild and Stiglitz (1970), Ekern (1980) characterized Nth-degree risk increases by the fact that they increase the ex-pected value of any N times continuously di¤erentiable real valued function q such that ( 1)Nq(N ) 0 where q(N ) is the Nth derivative of q: We prove a
dual result. The functions q for which the expected value is increased by any increase in Nth-degree risk are exactly those such that ( 1)Nq(N ) 0. This general result is the main ingredient for characterizing necessary and su¢ cient conditions for unambiguous comparative statics of risk within our framework and also for the interpretation of these results.
In Section 3 we perform the comparative statics analysis within the men-tioned framework and for two di¤erent scenarios, one in which the risk is additive and another one in which the risk is multiplicative in the decision variable. In each case, we provide necessary and su¢ cient conditions for an unambiguous impact of Nth degree risk increases and we compare our conditions with existing conditions in the literature. As an illustration, we show that no decision-maker that views the attributes as goods (i.e. with a positive marginal utility) will always increase the level of the decision variable when faced with a …rst-degree multiplicative-risk increase. Similarly, no decision-maker with diminishing mar-ginal utility (i.e. risk averse) will always increase the level of the decision variable when faced with a Rothschild-Stiglitz multiplicative-risk increase. While these results are intuitive in the classical portfolio choice problem, we show that they hold much more generally within our framework and that they generalize to increases in Nth-degree risk.
Section 4 develops a simple and intuitive interpretation of the previously obtained conditions. Following along the lines of Eeckhoudt and Schlesinger (2006), Eeckhoudt et al (2007), Eeckhoudt et al (2009), and Chiu et al. (2011), we propose concepts of additive and multiplicative directional Nth-degree risk aversion; these are characterized via preferences for harms disaggregation across outcomes of 50-50 bivariate lotteries. The harms we consider involve unidimen-sional increases in Nth-degree risk. These concepts of directional Nth-degree risk aversion include as special cases the concept of prudence analyzed by Eeck-houdt and Schlesinger (2006) and the concept of cross-prudence analyzed by Eeckhoudt et al (2007). For a …xed level of one of the attributes, the lotteries we study are isomorphic to a set of lotteries studied by Eeckhoudt et al (2009)
special case of this framework in which the attributes are perfect substitutes, making it, in essence, a univariate problem. Chiu et al (2011) study a broader class of problems that involve multiplicative risks in a univariate setting.
in the case of additive risks and to the lotteries studied by Chiu et al (2011) in the case of multiplicative risks. In addition to extending their results to the bidimensional case, our contribution is to characterize the preference for harms disaggregation (i.e. to establish necessary and su¢ cient conditions). In partic-ular, we characterize the situations where Nth degree changes in risk have an unambiguous impact on the optimal decision in terms of Nth-degree preference for harms disaggregation.
2
A Preliminary Result
Let us start by revisiting the notion of increases in risk proposed by Ekern (1980).
De…nition 1 (Ekern) Let e1ande2 denote two random variables with values
in [0; B] : For i = 1; 2, we denote by Fe[1]
i the distribution functions and, for k = 1; 2; ::: we de…ne the functions Fe[k+1]
i on R+ by Fe[k+1] i (x) = Z x 0 Fe[k] i(t)dt for x 2 R+:
We say that e2 is an increase in Nth-degree risk over e1; and we denote it by
e2 <N e1; if Fe[N ]2 (x) Fe[N ]1 (x) for all x 2 [0; B] where the inequality is strict
for some x and Fe[k]2(B) = Fe[k]1(B) for k = 1; :::; N:
For example, an increase in 2nd-degree risk coincides with Rothschild and Stiglitz’(1970) mean preserving increase in risk, while an increase in 3rd-degree risk coincides with a mean and variance preserving increase in risk that Menezes et al. (1980) labeled ’increase in downside risk’. An increase in 4th-degree risk is equivalent to what Menezes and Wang (2005) call an ’increase in outer risk’. Ekern (1980) characterizes increases in Nth-degree risk: he establishes that e2 is an increase in Nth-degree risk over e1 (as de…ned above) if and only
if E [q(e2)] > E [q(e1)] for all N times continuously di¤erentiable real valued
function q such that ( 1)Nq(N ) > 0 where q(N ) = dNq
d N: In particular, this means that Ekern (1980) shows that if a function q is such that ( 1)Nq(N ) >
0 then E [q(e2)] > E [q(e1)] for all pair (e1;e2) such that e2 <N e1: The
following Lemma extends this result by also showing the reverse implication. We characterize the set of N times continuously di¤erentiable functions for which E [q(e2)] E [q(e1)] for all pair (e1;e2) where e2is an increase in
Nth-degree risk overe1and we show that these functions are exactly those such that
( 1)Nq(N ) 0:
Lemma 2 Let q be a given real valued function that is N times continuously di¤ erentiable on R+. The following are equivalent.
2. For all x 0; we have ( 1)Nq(N )(x) 0:
As an illustration, for N = 2; we retrieve the classical result that a risk averse agent (i.e. an agent who dislikes mean preserving spreads) is an agent whose utility function is concave.
Lemma 2 will prove to be essential in Sections 3 and 4.
3
Optimal Decision and Increasing Risk
The problem that we analyze follows closely the setup in Dardanoni (1988). The decision-maker has an increasing, strictly concave and in…nitely di¤eren-tiable two-dimensional utility function U (y; z) de…ned for y and z positive. As usual, we denote partial derivatives by subscripts. Uncertainty is described by a probability space ( ; F; P ) where describes the set of possible states of the world, F is the set of measurable events and P is a probability measure. The decision maker has the possibility to buy a quantity x of an asset that has a random payo¤ ~ 0 in terms of the second attribute at a (deterministic) cost p > 0 in terms of the …rst attribute. The initial endowment of the decision maker in terms of the …rst attribute is deterministic and denoted by K: The initial endowment in terms of the second attribute is random and denoted by ~ 0: Both random variables ~ and ~ are assumed to be bounded above. The decision-maker’s problem PU;K;p(~; ~) is then the following
M axxE [U (K xp; x~ + ~)] : (1)
The …rst-order necessary condition and the second-order su¢ cient condition for optimality, assuming an interior solution, are given by
E [g (x; ~; ~)] = 0 (2)
E [gx(x; ~; ~)] < 0 (3)
with g (x; ; ) = pUy(K xp; x + ) + Uz(K xp; x + ) and where
gx(x; ; ) = p2Uyy(K xp; x + ) 2p Uyz(K xp; x + )+ 2Uzz(K xp; x + ) :
By strict concavity of U; this last quantity is always negative. The …rst order condition is then necessary and su¢ cient.
In the next we will alternatively need the following Inada-type conditions: Assumption A1 The utility function U is such that limz!0UUzy(y;z)(y;z) = 1 and limz!1UUzy(y;z)(y;z) = 0.
Assumption A2The utility function U is such that zUz(y;z)
Uy(y;z) is unbounded. It is classical to assume that limz!0Uz(y; z) = 1 and limz!1Uz(y; z) = 0
which means that the second attribute is necessary for the agent’s survival and that the agents approach satiation for large quantities of that attribute. As-sumption A1 is a little bit stronger and limits the possibilities for substitutabil-ity between the two attributes. A low level of the second attribute increases the
marginal utility for that attribute much faster than for the other one and satia-tion with respect to the second attribute does not mean satiasatia-tion with respect to the …rst one. An example of utility functions satisfying our Inada-type condi-tion is given by the set of homothetic utility funccondi-tions that satisfy the classical Inada conditions in the direction of both attributes, i.e. limz!0Uz(y; z) = 1,
limz!1Uz(y; z) = 0, limy!0Uy(y; z) = 1 and limy!1Uy(y; z) = 0. Clearly,
with a separable utility function U of the form U (y; z) = u(y) + v(z);, Assump-tion A1 is also equivalent to the classical Inada condiAssump-tion on v. AssumpAssump-tion A2 means that U must satisfy either limz!0zUUzy(y;z)(y;z) = 1 (which is a stronger condition than the …rst condition in Assumption A1) or limz!1zUUzy(y;z)(y;z) = 1. With a separable utility function; the condition can be rewritten as zv0(z) is
un-bounded. This last condition is, in particular, satis…ed by all CRRA functions except the logarithmic utility function. When U is nonseparable, our assump-tion is, in particular, satis…ed when zUz(y; z) is unbounded and when Uy(y; z)
remains bounded for z near to 0 or near to 1 (or converges to 1 slower than Uz(y; z)). Note that Assumption A2 is satis…ed when we embed the classical
portfolio problem with one risk-free asset and one risky asset in our framework (setting U (y; z) = v(y + z) and rede…ning the variables).
The problem PU;K;phas found many important applications in the economics
literature, including
The 2-date optimal saving model with either time-non-separable utility (e.g. Leland, 1968, Sandmo, 1970, Dreze and Modigliani, 1972) or time-separable utility (e.g. Kimball, 1990, Eeckhoudt and Schlesinger, 2008, Chiu et al., 2011) and uncertainty surrounding the rate-of-return on saving or the future income.
The optimal allocation of income to medical expenditures and consump-tion of non-medical goods when either the return on medical expenditures or the consumer’s health status is uncertain (e.g. Dardanoni and Wagsta¤, 1990).
The trade-o¤ between leisure and consumption, with wage income or non-wage income risks (e.g. Block and Heineke, 1973, Tressler and Menezes, 1980, Chiu and Eeckhoudt, 2010).
The trade-o¤ between (dirty) production and environmental-quality, with uncertainty surrounding the damages that the productive activity gener-ates or the level of environmental quality itself (e.g. Baiardi and Menegatti, 2011).
The private provision of public goods under uncertainty surrounding the contributions of others (Sandler et al., 1987, Keenan et al., 2006). The leisure/production trade-o¤ of an entrepreneur facing price uncer-tainty in a competitive environment (Rothschild and Stiglitz, 1971, Chiu et al., 2011).
In some of the above cases, like in the 2-date optimal saving model, the asset is a classical …nancial asset. In other cases, the asset corresponds to the mechanism that transforms money into health (health care) or leisure into money (labor) or production into environmental quality, etc.
The objective in these articles has been the evaluation of how increases in risk a¤ect the optimal value of the choice variable. While the earlier literature focused exclusively on mean preserving spreads, a number of recent papers have evaluated more generally the e¤ect of increases in Nth-degree risk as de…ned by Ekern (1980) (e.g. Eeckhoudt and Schlesinger, 2008, Chiu and Eeckhoudt, 2010, Baiardi and Menegatti, 2011, Chiu et al., 2011). In the present paper, we will also focus on the more general case of N th-degree risk, considering separately the problem in which the endowment of the second attribute ~ is random and the problem in which the asset’s payo¤ ~ is random.
3.1
Uncertainty Over the Endowment
Suppose that e = is deterministic and remains unchanged while considering a change in the second attribute initial endowment from e1 to e2, where e2
is an increase in Nth-degree risk over e1: How does the increase in risk from
e1 to e2 a¤ect the optimal level of the choice variable x? In order to provide
some intuition, let us consider the classical 2-date precautionary saving problem with a separable utility function U of the form U (y; z) = u(y) + v(z): In such a setting, it is well known that an agent raises his optimal saving when adding a zero-mean risk (to a deterministic second period initial endowment) if and only if his marginal utility of future consumption is convex, v000 0 (see e.g. Gollier, 2001). In our more general setting where U is not separable and where the initial endowment e1 is not necessarily deterministic, the analysis should
lead us to introduce conditions that involve cross derivatives as well as the cost
p of the second attribute in terms of the …rst attribute in order to re‡ect the
trade-o¤ between these two attributes. The following proposition provides a characterization of the utility functions for which any increase in Nth-degree risk increases the optimal level of the choice variable.
Proposition 3 Let U be a given increasing, strictly concave and in…nitely dif-ferentiable utility function satisfying Assumption A1. Let us consider p and as given. The following properties are equivalent:
1. For all initial endowment (K;e1); any increase in Nth-degree risk over the
second attribute initial endowment from e1 to e2 increases the optimal
level of the choice variable, i.e. x2 x1 where x1 and x2 respectively denote the solutions of PU;K;p(e1; ) and PU;K;p(e2; ) and wheree2<N
e1.
2. For all (y; z) ; we have ( 1)N +1p@@y@zN +1NU (y; z) + ( 1)
N @N +1U
@zN +1 (y; z) 0. Therefore, for example, fore1= k12 R+, ife2is an increase in …rst-degree
then x2 x1 (remark that, by concavity of U; Uzz is nonpositive). Similarly,
fore1= k12 R+, if e2is an increase in second-degree risk over e1of the form
e2 = k1+e", where e" is a mean zero random variable, and if Uyzz 0 and
Uzzz 0 then x2 x1. As a third example, if e1is a lottery with equiprobable
outcome k and 2k +e" (where k is a positive constant and e" is a mean zero random variable), if e2 is also a lottery with equiprobable outcomes 2k and
k +e", and if Uyzzz 0 and Uzzzz 0 then x2 x1.
If the problem under consideration corresponds to the classical 2-date pre-cautionary saving problem with endowment risk and with a separable utility function U of the form U (y; z) = u(y) + v(z), we obtain that x1 x2 for all pair (e1;e2) such that e14N e2if and only if ( 1)Nv(N +1) 0: The "if" part
of this result has been established by Eeckhoudt and Schlesinger (2008). We emphasize that our condition ( 1)Nv(N +1) 0 is necessary and su¢ cient.2
3.2
Uncertainty Over the Asset’s payo¤
Suppose now that e = is deterministic and remains unchanged while consid-ering a change frome1 to e2, where e2 is an increase in Nth-degree risk over
e1. The following Proposition establishes the analog of Proposition 3 for an
increase in risk of this multiplicative variable.
Proposition 4 Let U be a given increasing, strictly concave and in…nitely dif-ferentiable utility function satisfying Assumption A2. The following properties are equivalent:
1. For all initial endowment (K; ) and all asset’s cost and payo¤ (p;e1) such
that the solution x1 of PU;K;p( ;e1) is nonnegative, any increase in
Nth-degree risk over the asset’s payo¤ frome1toe2increases the optimal level
of the choice variable, i.e. x2 x1where x1 0 and x2respectively denote the solutions of PU;K;p( ;e1) and PU;K;p( ;e2) and where e2<N e1.
2. For all (y; z) ; we have ( 1)N @@y@zN +1NU(y; z) 0, ( 1)
N
z@@zN +1N +1U (y; z) + N
@NU
@zN (y; z) 0 and ( 1)N @@zNNU(y; z) 0.
The necessary and su¢ cient conditions in Proposition 4 appear to be sim-ilar to a number of results established previously in the context of di¤erent applications of our model3. There is, however, a crucial di¤erence. Consider
again the 2-date saving problem with separable utility, but now with rate-of-return risk. This problem was analyzed by Rothschild and Stiglitz (1971) in the context of increases in 2nd-degree risk and more recently by Eeckhoudt and Schlesinger (2008) for more general increases in risk. In both papers, the
2Similarly, in the context of problems with non-separable utility and an additive risk, Chiu
and Eeckhoudt (2010) and Baiardi and Menegatti (2011) established the su¢ ciency of the condition in Proposition 3.
3For example, Rothschild and Stiglitz (1971), Keenan et al. (2006), Eeckhoudt and
Schlesinger (2008), Chiu and Eeckhoudt (2010), Baiardi and Menegatti (2011), and Chiu et al. (2011).
authors show that when future labor income is zero (in our setting = 0, so z = x ) the consumer will save more in response to an increase in 2nd-degree risk if zv(3)(z) + 2v(2)(z) 0. According to our proposition, this condition is
indeed necessary and, as our proof makes it clear, it is also su¢ cient if = 0. But Proposition 4 also shows that the assumption of zero labor income does not come without loss of generality. Once we consider the more general case, another necessary condition for an increase in savings is v(2)(z) 0. In other
words, any risk-averse consumer (v(2)(z) < 0) will decrease savings in response
to higher risk for some initial endowment levels. In essence, Proposition 4 corre-sponds to Rothschild and Stiglitz’(1971, p. 72) conclusion that "no risk averse investor will always increase his holdings of risky assets when their variability increases." Proposition 4 implies that this important result holds much more generally in the context of problem PU;K;p( ; ~) and generalizes the conclusion
by stating that no agent for which ( 1)N @@zNNU (y; z) < 0 will always increase the demand for the asset in problem PU;K;p( ; ~) when facing an increase in
Nth-degree risk.
As another illustration, consider the static labor supply problem analyzed by Chiu and Eeckhoudt (2010). In this context y represents leisure, the choice vari-able x is labor supply, represents wage income, and represents non-wage in-come. Chiu and Eeckhoudt (2010) show that the conditions ( 1)N @@y@zN +1NU (y; x + ) 0 and ( 1)N x @@zN +1N +1U (y; x + ) + N@
N
U
@zN (y; x + ) 0 are su¢ cient for an increase in Nth-degree risk in wage income to increase the supply of labor.4
Note that the second condition does not correspond to our second condition in Proposition 4. Indeed, the derivatives are not taken at the same point and the two conditions coincide only for = 0: In fact, if we divide by @@zNNU (y; x + ) the condition introduced by Chiu and Eeckhoudt (2010) we obtain a condition on the concept of proportional N-th degree relative risk aversion 5. Instead, our condition relies on the concept of N-th degree relative risk aversion, which is more natural and easier to interpret. Our Proposition implies, in particular, that imposing the Chiu and Eeckhoudt (2010) condition on the proportional N-th degree relative risk aversion for all is equivalent to imposing a condition on the usual concept of N-th degree relative risk aversion as well as imposing ( 1)N @@zNNU(y; z) 0: This means that Proposition 2 of Chiu and Eeckhoudt (2010) cannot be applied in order to characterize the situations where the sup-ply of labor is increased in response to a risk increase at all initial endowment levels . Indeed, the authors assume that ( 1)N @@zNNU (y; z) 0, which is not consistent with the necessary condition ( 1)N @NU
@zN (y; z) 0. This implies, for instance, that a mean preserving spread in wage income cannot always in-crease the supply of labor for a consumer with diminishing marginal utility of
4Similar conditions, but in di¤erent contexts, have been derived by Baiardi and Menegatti
(2010) and by Chiu et al. (2011).
5We recall that the concept of proportional relative risk aversion is de…ned to be
zUzz(y;z+ )
Uz(y;z+ ) and the concept of N-th degree relative risk aversion is de…ned to be
consumption.
These results then suggest that, when considering a risk increase on the asset’s payo¤, it may be more natural to analyze the conditions under which the agent decreases his level of exposure to the asset’s risk. While Proposition 4 characterized the situations where any increase in risk on the asset’s payo¤ implies an increase in the choice variable, the following Proposition characterizes the situations where any increase in risk on the asset’s payo¤ implies a decrease in the choice variable.
Proposition 5 Let U be a given increasing, strictly concave and in…nitely dif-ferentiable utility function satisfying Assumption A2. The following properties are equivalent:
1. For all initial endowment (K; ) and all asset’s cost and payo¤ (p;e1)
such that the solution x1 of PU;K;p( ;e1) is nonnegative, any increase in
Nth-degree risk over the asset’s payo¤ frome1 toe2 decreases the optimal level of the choice variable, i.e. x1 x2 where x1 0 and x2 respectively denote the solutions of PU;K;p( ;e1) and PU;K;p( ;e2).
2. For all (y; z) ; we have ( 1)N @@y@zN +1NU(y; z) 0, ( 1)
N
z@@zN +1N +1U (y; z) + N@ NU
@zN (y; z) 0 and ( 1)N @@zNNU(y; z) 0.
For example, if we consider the 2-date optimal saving problem with rate-of-return risk, we obtain that any increase in N-th degree risk leads to a decrease of the optimal saving (i.e. x2 x1 for all e1 4N e2) if and only if, for all z;
( 1)Nv(N )(z) 0 and ( 1)N
zv(N +1)(z) + N v(N )(z) 0, which corresponds
to the conditions established by Eeckhoudt and Schlesinger (2008) in a model without labor income ( = 0).6 Again, we remark that our conditions are
necessary and jointly su¢ cient and that they hold for all (y; z). Still in a one dimensional setting, we retrieve the following well-known results in the context of the classical portfolio choice problem (see e.g. Rothschild and Stiglitz, 1971, Hadar and Seo, 1990, Gollier, 2001, p. 61):
A risk averse agent decreases his optimal demand for the risky asset at all wealth levels and when facing an increase in 1st-degree risk in the asset’s payo¤ if the degree of relative risk aversion is positive and no greater than one, 0 < zvv(2)(1)(z)(z) 1.
A risk averse agent decreases his optimal demand for the risky asset at all wealth levels and when facing an increase in 2nd-degree risk in the asset’s payo¤ if the degree of relative prudence is positive and no greater than two, 0 < zvv(3)(2)(z)(z) 2:
6Suppose, for example, that relative risk aversion is constant: v(z) = (1 ) 1z1 :
Then, if < 1an Nth-degree risk increase in the rate-of-return will decrease savings. For > 1;an Nth-degree risk increase in the rate-of-return will increase savings for some wealth levels and it will decrease savings for other wealth levels.
More generally, a direct implication of our results is that an agent will decrease his portfolio exposure to the risky asset whenever he faces an in-crease in Nth-degree risk in the asset’s payo¤ and his utility function satis…es 0 zv(N +1)v(N )(z)(z) N with ( 1)
N
v(N )(z) 0.
In summary, Propositions 3, 4, and 5 establish precisely the necessary and su¢ cient conditions for unambiguous comparative statics of changes in risk in a large class of problems. Our next objective is to develop a simple and intuitive approach to interpret these conditions.
4
Lottery Choices and Optimal Exposure to Risk
In this section we show that the optimal response to changes in risk can be characterized via preferences over particular classes of lottery pairs. We …rst consider the case of additive risks and then evaluate the case of multiplicative risks.
4.1
Additive Risks
Consider two attributes with nonnegative initial quantities y and z, and imagine a lottery in which with a 50 percent chance risk e1 is added to z and with a
50 percent chance risk e2 is added to z, where e2 is an increase in Nth-degree
risk overe1: Denote this lottery by [(y; z +e1) ; (y; z +e2)]. Now consider the
following location experiment: the consumer is told that she must accept the bundle x1 y; x1 z in tandem with one of the lottery outcomes of her choice and the bundle x2 y; x2 z in tandem with the other lottery outcome where x1; x2; y; z are constants and x2> x1; to which outcome will she a¤ect each
bundle?
The answer to the preceding question clearly depends not only on the con-sumer’s preferences but also on the magnitude and the direction of y; z .
Let us …rst consider 1st-degree and 2nd-degree increases in risk. In par-ticular, let (e1;e2) = (k; 0) with k > 0, for a 1st-degree increase in risk and
(e1;e2) = (k; k +e"), where e" is a mean zero random variable bounded below by
k, for a 2nd-degree increase in risk. We propose the following de…nitions. De…nition 6 Let us consider a consumer endowed with preferences over the space of bivariate 50-50 lotteries denoted by :
1. We say that preferences display risk aversion in the direction of y; z if, for any (y; z; x1; x2) 2 R4+ with x2> x1, the consumer prefers to a¤ ect
the bundle x1 y; x1 z to the highest outcome
y + x2 y; z + x2 z ; y + x1 y; z + x1 z+ k
2. We say that preferences display prudence (or downside risk aversion) in the direction of y; z if, for any (y; z; x1; x2) 2 R4+ with x2 > x1, the
consumer prefers to locate the bundle x1 y; x1 z to the outcome of the lottery where the risk is not present
y + x2 y; z + x2 z+ k +e" ; y + x1 y; z + x1 z+ k
y + x1 y; z + x1 z+ k +e" ; y + x2 y; z + x2 z+ k : (5)
To understand these concepts it might be useful to consider a few special cases that were analyzed by Eeckhoudt and Schlesinger (2006) and Eeckhoudt et al. (2007)7. Suppose …rst that
y= 0 and z> 0, in which case, our
de…ni-tions coincide with the univariate de…nition of risk aversion and of prudence in attribute z as in Eeckhoudt and Schlesinger(2006). Next, suppose that y> 0 and z= 0, we recover Eeckhoudt et al.’s (2007) de…nitions of correlation aver-sion and of cross-prudence. As explained by Eeckhoudt et al. (2007), in all of these cases the consumer views the risks (either 0 instead or k together with x1 y; x1 z ; or k +e" instead of k together with x1 y; x1 z ) as ’mutually
aggravating,’so she prefers to disaggregate the "harms" across outcomes of the lotteries (notice that we are referring to x1 y; x1 z as a ’harm’ relative to
x2 y; x2 z ).
In essence, the same intuition holds for our more general de…nitions. When both y and z are positive, our de…nitions intuitively correspond to the
stan-dard notions of risk aversion and prudence, in the sense that a consumer who views risks as mutually aggravating would prefer to add the "bad" bundle x1 y; x1 z to the less risky outcome of the lottery. If, on the other hand,
y and z have opposite signs, we would expect a consumer who views risks as
mutually aggravating to evaluate the relative strength of two e¤ects, one that aggravates the existing risk and the other one that mitigates it. Our bivariate de…nitions of directional risk aversion and of directional prudence cover all of these di¤erent cases.
In an expected utility framework, when preferences are represented by a bivariate utility function U (y; z) on R+ R+, they display risk aversion in the
direction of y; z if 1 2U y + x2 y; z + x2 z + 1 2U y + x1 y; z + x1 z+ k > 1 2U y + x1 y; z + x1 z + 1 2U y + x2 y; z + x2 z+ k : (6) The preferences display prudence in the direction of y; z if
7Eeckhoudt et al. (2007) restrict themselves to the case x
1 = 0. All our results could
be generalized further, along the lines of Eeckhoudt et al’s (2009) results in a univariate framework, by considering the case in which x1 is a …rst order stochastically dominated shift
1 2E U y + x2 y; z + x2 z+ k +e" + 1 2 U y + x1 y; z + x1 z+ k > 1 2E U y + x1 y; z + x1 z+ k +e" + 1 2 U y + x2 y; z + x2 z+ k (7): We obtain the following result
Proposition 7 Let U be a given increasing, strictly concave and in…nitely dif-ferentiable utility function on R2
+. The preferences represented by U display
risk aversion in the direction of y; z if and only if yUyz+ zUzz < 0 for
all (y; z) 2 R2
+. Preferences display prudence in the direction of y; z if and
only if yUyzz+ zUzzz > 0 for all y and z.
There are, in fact, two e¤ects that determine the consumer’s bivariate pref-erence for ’harm disaggregation’. The condition for risk aversion in the direction of y; z can be decomposed into 1) a ’risk aversion in z’e¤ect, captured by the term zUzz, and 2) a ’correlation aversion’ e¤ect, captured by the term yUyz. Similarly, the condition for prudence in the direction of y; z can be
decomposed into 1) a ’prudence in z’e¤ect, captured by the term zUzzz, and
2) a ’cross-prudence’e¤ect, captured by the term yUyzz.
These results can be quite easily generalized to the case of increases in Nth-degree risk. To do so, let us introduce the following de…nition.
De…nition 8 We say that preferences display Nth-degree risk aversion in the direction of y; z if, for all (y; z; x1; x2) 2 R4+ such that x2 > x1 and for all
pair of random variables (e1;e2) such that e2is an increase in Nth-degree risk
overe1; we have
y + x2 y; z + x2 z+e2 ; y + x1 y; z + x1 z+e1
y + x1 y; z + x1 z+e2 ; y + x2 y; z + x2 z+e1 : (8)
Again, this lottery ordering captures a preference to ’disaggregate harms’ across lottery outcomes, where the harms are represented by an Nth-degree risk increase (e2 <N e1) and by a shift from x2 to x1 < x2 in the direction
of y; z . The following proposition establishes precisely what this lottery preference means in an expected utility framework.
Proposition 9 Let U be a given increasing, strictly concave and in…nitely dif-ferentiable utility function on R2
+. The preferences represented by U display
Nth-degree risk aversion in the direction of y; z if and only if ( 1)Nfz(N ) 0
where f is de…ned on R2
+ by f (y; z) = yUy(y; z) + zUz(y; z):
In the case with y = 0 and z > 0, and for a …xed value of y, Eeckhoudt et al. (2009) established the “if” part of this Proposition.8 The contribution
8Eeckhoudt et al. (2009) showed more generally that if e
2 <N e1 and ~x1 <M ~x2, then
the 50-50 lottery [e2+ ~x1, e1+ ~x2] is an (N+M)th-degree risk increase over [e2+ ~x2, e1+
~
x1] in the sense of Ekern (1980), i.e. the second lottery is preferred by all decision makers
with ( 1)N +M @N +MU
@zN +M (y; z) 0. When y= 0and z > 0, and for a …xed value of y, our
of our proposition is twofold. First, and most obvious, our bivariate notion of directional Nth-degree risk aversion is more encompassing. Second, by making use of Lemma 2, the proposition characterizes a unique set of expected utility maximizers that display Nth-degree risk aversion in the direction of y; z . This is important because, without the “only if” part of the proposition, we would not be able to establish a direct link between lottery choices and optimal exposure to risk, as we do in the following corollary.
Corollary 10 Let U be a given increasing, strictly concave, in…nitely di¤ er-entiable utility function on R2
+, satisfying Assumption A1. Let us consider
(p; ) 2 R2
+ as given. The following properties are equivalent:
1. For all initial endowment (K;e1); any increase in Nth-degree risk over the
second attribute initial endowment increases the optimal level of the choice variable, i.e. x2 x1 where x1and x2respectively denote the solutions of PU;K;p(e1; ) and PU;K;p(e2; ) and wheree2<N e1.
2. The preferences represented by U display Nth-degree risk aversion in the direction of ( p; ).
This result is quite intuitive. It establishes that, in response to an increase in risk, the consumer’s optimal level of the choice variable (e.g. the level of savings, of labor supply, of medical care) will re‡ect her preferences towards ’harms disaggregation’as de…ned in this paper. Consider, for instance, the case of a mean preserving spread and, to be concrete, the problem of precautionary saving with time-non-separable utility. As explained above, two e¤ects operate. First, a consumer that is prudent (in second period consumption) would like to allocate a higher level of savings to second period consumption to mitigate the increase in risk. Second, a higher level of savings implies a lower level of …rst period consumption, which a cross-prudent consumer dislikes to match with the higher risk. As a result, the higher level of risk implies a higher level of savings if the prudence e¤ect is stronger than the cross-prudence e¤ect, i.e. pUyzz+ Uzzz > 0, or, equivalently, if the consumer is prudent in the direction
of ( p; ). Similarly, consider a decrease in second period income with certainty (e2is then a …rst degree increase in risk overe1). A consumer that is risk averse
(in second period consumption) would like to mitigate this harm by increasing savings. Doing so, however, implies that the harm will be present when …rst period consumption is lower, which a correlation averse individual dislikes. As a result, this harm will increase savings if the risk aversion e¤ect is stronger than the correlation aversion e¤ect, i.e. pUyz + Uzz < 0, or, equivalently, if the
consumer is risk averse in the direction of ( p; ).
Clearly, a similar intuition holds for all of the above-mentioned applications and for increases in risk of any degree. A decision-maker that views risks as mutually aggravating would like to compensate the higher risk in attribute z with a higher level of the choice variable. Alternatively, he could compensate the higher risk by reducing the level of the choice variable and, as a result, by increasing the level of the other attribute. Whether the level of the choice
variable increases or decreases in response to an increase in risk then depends on the relative strengths of two opposing forces, one that aggravates the higher risk and the other one that ameliorates it. Our contribution in this section has been to establish the equivalence between such a trade-o¤ and the choice of simple bivariate lotteries.
4.2
Multiplicative Risks
As before, consider the following experiment: the consumer needs to locate x1 y; x1 z and x2 y; x2 z to each of the two (random) outcomes of the
lottery (y; z +e1) and (y; z +e2) where x2 > x1 0 and where e2 0 is an
Nth degree risk increase over e1 0. Now, however, the e¤ect of x1 z and
x2 z is to scale the risks e1 and e2: In particular, the consumer evaluates the
following lotteries:
L1 = y + x2 y; z + x2 ze2 ; y + x1 y; z + x1 ze1 (9)
L2 = y + x1 y; z + x1 ze2 ; y + x2 y; z + x2 ze1 :
We propose the following de…nitions.
De…nition 11 We say that preferences display Nth-degree multiplicative-risk attraction (resp aversion) in the direction of y; z if, for all (y; z; x1; x2)
such that x2 > x1 0 and for all pair of random variables (e1;e2) such that
e2 is an increase in Nth degree risk over e1; we have L1 L2(resp. L2 L1).
To understand the di¤erent forces at play in such preference ordering, con-sider …rst the case with y = 0 and a …xed value of y analyzed by Chiu et al. (2011). On the one hand, since x2 > x1, the higher risk in L1 is scaled up,
which hurts a consumer that dislikes higher risks. On the other hand, x2> x1
also implies that the distribution of z + x2 ze2 is shifted upwards relative to
z+x1 ze1: As in the previous section, this implies that an individual that prefers
to disaggregate harms would like to match this higher level of z with the higher risk, making L1relatively more attractive than L2. Therefore, as stated by Chiu
et al. (2011), the choice of L1 over L2 will depend on the relative strengths of
these two opposing e¤ects. If we now allow y to di¤er from zero we have, as in the previous section, another e¤ect that arises from the consumer’s preference to match the higher risk with a higher level of the other attribute (y). If y> 0 this additional e¤ect will make L1more desirable than L2for an individual that
prefers to disaggregate harms, while the opposite will be true if y < 0. In the next, we will focus on the case y 0 and this tends to make L2more desirable
than L1 for an individual that prefers to disaggregate harms.9 It is in reference
to this situation that we choose to call Nth-degree risk aversion the fact that L2
is preferred to L1and to call Nth-degree risk-attraction the opposite behavior.
9The case with
y 0and z 0is the relevant scenario to interpret the conditions found
The following Proposition establishes precisely the di¤erent forces at play in an expected utility model and for general increases in Nth-degree risk.
Proposition 12 Let U be a given increasing, strictly concave and in…nitely dif-ferentiable utility function on R2
+ and let y 0 and z 0 be given. The
pref-erences represented by U display Nth-degree multiplicative-risk attraction (resp. aversion) in the direction of y; z if and only if ( 1)N @@y@zN +1NU(y; z) 0 (resp.
0); ( 1)N @@zNNU(y; z) 0 (resp. 0) and ( 1)
N
z@@zN +1N +1U(y; z) + N@ NU
@zN(y; z) 0 (resp. 0):
Note that the characterizations do not depend on y; z but only on their
signs since we assumed y 0 and z 0: In the next we will say that
the preferences represented by U display Nth-degree multiplicative-risk aver-sion (attraction) in the direction of R R+ in order to say that they
dis-play Nth-degree multiplicative-risk aversion (attraction) in the direction of some
y; z 2 R R+ or equivalently in the direction of all y; z 2 R R+:
We have then the following immediate corollaries.
Corollary 13 Let U be a given increasing, strictly concave and in…nitely dif-ferentiable utility function satisfying Assumption A2. The following properties are equivalent:
1. For all initial endowment (K; ) and all asset’s cost and payo¤ (p;e1)
such that the solution x1 of PU;K;p( ;e1) is nonnegative, an increase in
Nth-degree risk over the asset’s payo¤ , from e1 to e2, increases (resp.
decreases) the optimal level of the choice variable, i.e. x2 x1 (resp. x1 x2), where x1 0 and x2 0 respectively denote the solutions of PU;K;p( ;e1) and PU;K;p( ;e2).
2. The preferences represented by U display Nth-degree multiplicative-risk at-traction (resp. aversion) in the direction of R R+:
In other words, an individual will always decrease the demand for the asset in problem PU;K;p( ;e) when the asset’s payo¤ is subject to an increase in
Nth-degree risk if and only if he or she always selects lottery L2 over lottery
L110. Such a choice re‡ects, in turn, the decision-maker’s attitudes towards
disaggregation of harms across bidimensional lottery outcomes as well as the decision-maker’s attitudes towards increases in risk. Similarly, an individual will always increase the demand for the asset in problem PU;K;p( ;e) when the
asset’s payo¤ is subject to an increase in Nth-degree risk if and only if he or she always selects lottery L1over lottery L2. However, as it is clear from Proposition
12, no individual that displays mixed risk aversion in z, ( 1)N @@zNNU(y; z) 0, can also display Nth-degree multiplicative-risk attraction in the direction of R R+, so no such individual will always prefer lottery L1over lottery L2, or
1 0We emphasize again that the "only if" part of Proposition 12 is the critical ingredient to
equivalently, no such individual will always increase the demand for the asset in problem PU;K;p( ;e).
Remark. As mentioned above, Chiu et al. (2011) analyzed the univariate
case with y = 0 and y …xed. They conclude that (Theorem 2): (using our
notation) if ( 1)n @@znUn(y; z) 0 for n = N; N + 1, then L1 ( ) L2if and only
if ( 1)N x@@zN +1N +1U(y; x + z) + N
@NU
@zN(y; x + z) ( ) 0 for all x 0. As it is clear from the proof of Proposition 12, the second part of this statement is equiv-alent to our results when y = 0. We remark that the "only if" part follows from
our Lemma 2, which to our knowledge has not been proven before. Furthermore, our proof also clari…es that the assumption ( 1)N @NU
@zN(y; z) 0 is not consis-tent with the condition ( 1)N x@N +1U
@zN +1(y; x + z) + N@ N
U
@zN(y; x + z) 0. In fact, as stated in Proposition 12, the condition ( 1)N @@zNNU(y; z) 0 is necessary for an individual to always prefer L1over L2in an expected utility framework.
5
Concluding Remarks
Given the ubiquitous presence of uncertainty in most economic decisions, it is not surprising that a large amount of research has been devoted towards under-standing the economic consequences of changes in risk. This paper contributes towards that goal in two important ways. First, we establish the minimum set of necessary and su¢ cient conditions for unambiguous comparative statics of changes in risk in the large class of problems involving bidimensional conse-quences. Second, we link these conditions with more primitive attitudes towards risk in the form of preferences over simple lottery pairs. In particular, we show that making unambiguous statements about the ordering of a particular class of lottery pairs is equivalent to making unambiguous statements regarding the optimal response to changes in risk in the problems under consideration.
Two interesting and related topics for future research are 1) establishing measures of the intensity of preferences towards higher-order risks in a multidi-mensional setting and 2) evaluating comparative statics of risk when the risk is multidimensional.
6
Appendix
Proof of Lemma 2. The fact that 2. implies 1. results directly from Ekern (1980). For the sake of completeness we rederive it. Let (e1;e2) be such that
e2<N e1: We have E [q(e2)] E [q(e1)] = Z B 0 q(x)dFe2(x) Z B 0 q(x)dFe1(x) = N X k=1 ( 1)k 1q(k 1)(B) h Fe[k] 2(B) F [k] e1(B) i + Z B 0 ( 1)Nq(N )(x)hFe[N ] 2 (x) F [N ] e1 (x) i dx: By de…nition, all the terms in the sum are equal to 0 and Fe[N ]2 (x) Fe[N ]1 (x) 0 on [0; B] : By 2., the integral is then nonnegative and E [q(e2)] E [q(e1)].
Proof. Let us now prove that 1. implies 2. Let q be an N times continuously di¤erentiable function on R+such that E [q(e2)] E [q(e1)] for all pair (e1;e2)
such thate2<N e1: Let, if it exists, be a nonnegative real number such that
q(N )( ) 6= 0 and let " denote a positive real number such that q(N )(t) 6= 0 for t 2 [ ; + "] : The sign of q(N )remains then constant on [ ; + "] : Let e1and
e2 be two nonnegative bounded above random variables such that e2 <N e1:
Let B be a common upper bound for e1 and e2 and let e1 = B"e1+ and
e2 = B"e2+ : The random variables e1 and e2 take their values in [ ; + "]
and it is easy to check that Fe[k]i (t) = B" k 1Fe[k] i
t
" B for k = 1; 2::: and
i = 1; 2. Therefore, e2<N e1 which implies, by 1., that E [q(e2)] E [q(e1)] :
We have E [q(e2)] E [q(e1)] = Z +" q(x)dFe2(x) Z +" q(x)dFe1(x) = N X k=1 ( 1)k 1q(k 1)( + ")hFe[k] 2( + ") F [k] e1( + ") i + Z +" ( 1)Nq(N )(t) h Fe[N ]2 (t) Fe[N ]1 (t) i dt: By de…nition, we have Fe[k] 2( + ") = F [k] e1( + ") for k = 1; :::; N and F [N ] e2 (t) Fe[N ]1 (t) 0 and the inequality is strict for some t in [ ; + "] ; and even on a neighborhood of t by continuity of Fe[N ]
1 and F
[N ]
e2 : Since the sign of q
(N )
remains constant on [ ; + "] ; this gives that ( 1)Nq(N )(t) > 0 on [ ; + "]
and ( 1)Nq(N )(x) > 0 for all x such that q(N )(x) 6= 0 which completes the
proof.
Proof of Proposition 3 Let us prove that 2. implies 1. Let (e1;e2) be
such that e2 <N e1 and let q( ) = g(x1; ; ): We have g (N ) (x1; ; ) = p@@y@zN +1NU (K x1p; x1 + )+ @ N +1U @zN +1 (K x1p; x1 + ) : By 2., we have ( 1) N g(N )(x1; ; ) 0 and ( 1)Nq(N )( ) 0: By Lemma 2, this leads to E [q(e
1)] E [q(e2)] : By
0: By concavity of U; it is easy to check that E [g(x;e2; )] is a decreasing
function of x: Since x2 is characterized by E [g(x2;e2; )] = 0; we obtain that
x2 x1:
Proof. Let us prove 1. implies 2. As in the proof of Lemma 2, we consider e1 and e2 two nonnegative random variables with a common upper bound B such that e2 <N e1: As above, we introduce the random variables e1;" =
"
Be1+ and e2;"= B"e2+ for some > 0 and some " > 0: The random
variables e1;" and e2;" take their values in [ ; + "] and e2;" <N e1;": Let
us now consider ( 0; F0; P0) a copy of ( ; F; P ): We consider the probability space ( [ 0; G; Q") where G = fA [ A0 : (A; A0) 2 F F0g and where Q" is
de…ned by Q"(A [ A0) = "P (A) + (1 ")P0(A0) for (A; A0) 2 F F0: Let
us consider a given random variable e on ( 0; F0; P0) and let us de…ne the
random variablesei;"; i = 1; 2; on [ 0byei;"(!) =ei;"(!)1!2 +e(!)1!2 0: Since we have to deal with di¤erent probabilities, we will specify the probability under which the distribution function is determined by a subscript, for instance Fe;P denotes the distribution function of e under P: It is easy to check that Fei;";Q" = "Fei;";P + (1 ")Fe;P0 and we have then e2;"<N e1;": Let x1;" and
x2;" be respectively the solutions of PU;K;p(e1;"; ) and PU;K;P(e2;"; ) where
the expectations are taken with respect to Q": By 1., we have x2;" x1;": By
de…nition, we have EQ" g(x
1;";e1;"; ) = 0 and EQ" g(x2;";e2;"; ) = 0: By
concavity of U; g is a decreasing function of x and we have EQ" g(x
1;";e2; )
0: Let q be de…ned by q( ) = g(x1;"; ; ): We have EQ" q(e 2;") EQ" q(e1;") = EP[q(e2;")] EP[q(e1;")] " = N X k=1 ( 1)k 1q(k 1)( + ") h Fe[k]2;P( + ") Fe[k]1;P( + ") i " +" Z +" ( 1)Nq(N )(t) h Fe[N ]2;P(t) Fe[N ]1;P(t) i dt: By construction, the left side of the equality is nonnegative, all the terms in the sum are equal to zero and Fe[N ]2;P(t) Fe[N ]1;P(t) is nonnegative and nonzero. There-fore, ( 1)Nq(N )(t) is nonnegative at least on a given subinterval of [ ; + "] :
Let " be in [ ; + "] such that ( 1)Nq(N )(
") 0: We have then ( 1)N +1p@ N +1U @y@zN K x1;"p; x1;" + " +( 1)N @N +1U @zN +1 K x1;"p; x1;" + " 0 (10) where x1;" satis…es EQ" pU y K x1;"p; x1;" +e1;" + Uz K x1;"p; x1;" +e1;" = 0 or "EP h g(x1;"; " Be1+ ; ) i + (1 ")EP0 g(x1;";e; ) = 0: (11) Remark that, until now, ; K ande have been arbitrarily chosen. Let us now choose them carefully in order to derive our result. Let (Y; Z) be arbitrary in
R+ 2
. By our Inada condition, limz!0UUzy(Y;z)(Y;z) = 1 and limz!1
Uz(Y;z)
Uy(Y;z) = 0 which gives that there exists some z > 0 such thatUz(Y;z )
Uy(Y;z ) =
p or pU
y(Y; z )+
Uz(Y; z ) = 0: We choose x > 0 such that x < inf(Z; z ) and and K
are taken such that = Z x > 0 and K = Y + px : Let be given by
= z x > 0 and let e be a random variable on 0 that is equal to ;
P0 a.s. We have EP0[g(x ;e; )] = EP0[ pU y(K x p; x +e) + Uz(K x p; x +e)] = pUy(K x p; x + ) + Uz(K x p; x + ) = pUy(Y; z ) + Uz(Y; z ) = 0:
The solution of Equation (11) for " = 0 is then given by x : In a well chosen neighborhood of x , x 7 ! K xp and x 7 ! x +B"e1+ are bounded and
bounded away from 0. The functions Uy and Uz being continuously
di¤eren-tiable, the function (x; ") 7 ! "EP h
g(x;B"e1+ ; ) i
+ (1 ")EP0[g(x;e; )] is then di¤erentiable with respect to x at (x ; 0): Furthermore, the derivative of this last function with respect to x at (x ; 0) is nonzero (concavity of U ). The solution x1;" of Equation (11) is then continuous with respect to " in a neighborhood of 0 which gives lim"!0x1;" = x : Furthermore, we clearly have
lim"!0 "= : Taking the limit in Equation (10) when " tends to 0; we obtain
( 1)N +1p@ N +1U @y@zN (K x p; x + )+( 1) N @N +1U @zN +1 (K x p; x + ) 0 or, by construction ( 1)N +1p@ N +1U @y@zN (Y; Z) + ( 1) N @N +1U @zN +1 (Y; Z) 0
Proof of Proposition 4. Let us prove that 2. implies 1. Let (e1;e2) be such
that e2<N e1 and let q( ) = g(x1; ; ): We have q(N )( ) = g (N ) (x1; ; ) = p (x1)N @N +1U @y@zN (K x1p; x1 + )+ (x1)N @ N +1 U @zN +1 (K x1p; x1 + )+N (x1) N 1 @N U @zN (K x1p; x1 + ).
Since we assumed that x1 0, by 2, we have ( 1) N p (x1)N @ N +1 U @y@zN (K x1p; x1 + ) 0 and ( 1)N x1 @ N +1U @zN +1 (K x1p; x1 + ) + N @NU @zN (K x1p; x1 + ) = ( 1)N x1 x1 + (x1 + ) @N +1U @zN +1 (K x1p; x1 + ) + N @NU @zN (K x1p; x1 + ) + ( 1)N N x1 + @NU @zN (K x1p; x1 + ) 0
which gives ( 1)Nq(N )( ) 0: By Lemma 2, this leads to E [q(e1)] E [q(e2)] :
By de…nition, we have E [q(e1)] = 0 which gives E [q(e2)] = E [g(x1; ;e2)] 0:
By concavity of U; it is easy to check that g(x; ; ) is a decreasing function of x: From there we derive that x2 x1:
Proof. Let us prove that 1. implies 2. As in the proof of Lemma 2 and
Proposition 3, we consider e1 and e2 two nonnegative random variables with a common upper bound B such that e2 <N e1: As above, we introduce the
random variablese1;"= B"e1+ ande2;"= B"e2+ for some > 0 and some
" > 0: The random variablese1;" and e2;" take their values in [ ; + "] and
e2;"<N e1;": Let us consider ( 0; F0; P0) a copy of ( ; F; P ) : We consider the
probability space ( [ 0; G; Q") where G = fA [ A0 : (A; A0) 2 F F0g and
where Q"is de…ned for (A; A0) 2 F F0 by Q"(A [A0) = "P (A) + (1 ")P0(A0):
Let us consider a given random variablee on ( 0; F0; P0) and let us de…ne the
random variablesei;"; i = 1; 2; on [ 0 byei;"(!) =ei;"(!)1!2 +e(!)1!2 0: As previously, we have Fei;";Q"= "Fei;";P + (1 ")Fe;P0 ande2;" <N e1;": Let x1;" and x2;" be respectively the solutions of PU;K;p( ;e1;") and PU;K;p( ;e2;")
where the expectations are taken with respect to Q": If x1;"is nonnegative then,
by 1., we have x2;" x1;": By de…nition, we have EQ" g(x
1;"; ;e1;") = 0 and
EQ" g(x
2;"; ;e2;") = 0: By concavity of U; g is a decreasing function of x and
we have EQ" g(x 1;"; ;e2;") 0: Let q be de…ned by q( ) = g(x1;"; ; ): We have EQ" q(e 2;") EQ" q(e1;") = EP q(e2;") EP q(e1;") " = N X k=1 ( 1)k 1q(k 1)( + ") h Fe[k] 2;P( + ") F [k] e1;P( + ") i " +" Z +" ( 1)Nq(N )(t) h Fe[N ] 2;P(t) F [N ] e1;P(t) i dt: By construction, the left side of the equality is nonnegative, all the terms in the sum are equal to zero and Fe[N ]
2;P(t) F
[N ]
e1;P(t) is nonnegative and nonzero. There-fore, ( 1)Nq(N ) is nonnegative at least on a given subinterval of [ ; + "] :
Let " be in [ ; + "] such that ( 1)Nq(N )(
") 0: We have then ( 1)N +1p x1;" N @N +1U @y@zN K x1;"p; x1;" "+ + ( 1) N " x1;" N @N +1U @zN +1 K x1;"p; x1;" "+(12) +( 1)NN x1;" N 1@ NU @zN K x1;"p; x1;" "+ (13) 0 (14) where x1;" satis…es EQ" pU y K x1;"p; x1;"e1;"+ +e1;"Uz K x1;"p; x1;"e1;"+ = 0 or "EP h g(x1;"; ; " Be1+ ) i + (1 ")EP0 g(x1;"; ;e) = 0: (15)
Remark that, until now p; ; K; ande have been arbitrarily chosen. Let us now choose them carefully in order to derive our result. We assume …rst that e is equal to 1 on 0; P0 a.s. Let x > 0 be given and let (M; Y; Z) in (R
+)3
such that M < Z: We take p = Uz(Y;x +Z M )
Uy(Y;x +Z M ), K = Y + px ; = Z M and =Mx : By construction, we have > 0 and we have
EP0[g(x ; ;e)] = EP0
[ pUy(K x p; x e + ) + eUz(K x p; x e + )]
= pUy(K x p; x + ) + Uz(K x p; x + )
= pUy(Y; x + Z M ) + Uz(Y; x + Z M )
= 0:
The solution of Equation (15) for " = 0 is then given by x : In a well chosen
neighborhood of x , x 7 ! K xp and x 7 ! x +B"e1 + are bounded
and bounded away from 0. The functions Uy and Uzbeing continuously
di¤er-entiable, the function (x; ") 7 ! "EPhg(x; ; "
Be1+ )
i
+(1 ")EP0
[g(x; ;e)] is then di¤erentiable with respect to x at (x ; 0): Furthermore, the deriva-tive with respect to x at (x ; 0) is nonzero. The solution x1;" of Equation (15) is then continuous with respect to " in a neighborhood of 0 which gives lim"!0x1;" = x and guarantees that x1;" > 0 for " small enough: Since we
clearly have lim"!0 "= ; taking the limit in Equation (12) when " tends to
0; we obtain ( 1)N(x )N 1 px @ N +1U @y@zN (Y; Z) + x @N +1U @zN +1 (Y; Z) + N @NU @zN (Y; Z) 0 or ( 1)N px @ N +1U @y@zN (Y; Z) + M @N +1U @zN +1 (Y; Z) + N @NU @zN (Y; Z) 0 (16)
This result being true for all (Y; Z) in R+ 2
; all M 2 (0; Z) all x > 0 and for p = Uz(Y;x +Z M )
Uy(Y;x +Z M ): When x goes to 0, Y; M and Z being …xed, x Uz(Y;x +Z M )
Uy(Y;x +Z M ) goes to 0 and we obtain that
( 1)N M@
N +1U
@zN +1 (Y; Z) + N
@NU
@zN (Y; Z) 0
for all (Y; Z) in R+ 2
and all M 2 (0; Z) or equivalently
( 1)N Z@ N +1U @zN +1 (Y; Z) + N @NU @zN (Y; Z) 0 and ( 1) N@NU
@zN (Y; Z) 0 for all (Y; Z) in R+ 2
: We assumed that zUz(Y;z)
Uy(Y;z) is unbounded. We then have either limz!0z
Uz(Y;z)
Uy(Y;z) = 1 or limz!1zUUzy(Y;z)(Y;z) = 1. if limz!0z
Uz(Y;z)
and M = Z 2for arbitrarily small to make the quantity px = Uz(Y;x +Z M )
Uy(Y;x +Z M )x =
1 2
Uz(Y; )
Uy(Y; ) arbitrarily large. Since M; Y and Z are bounded, Equation (16) gives then ( 1)N @@y@zN +1NU (Y; Z) 0. If limz!1zUUzy(Y;z)(Y;z) = 1; it su¢ ces to take
x su¢ ciently large to make the quantity x + Z M su¢ ciently large and
Uz(Y;x +Z M )
Uy(Y;x +Z M )(x + Z M ) arbitrarily large. Since Z is kept …xed,
x x +Z M is
arbitrarily close to 1 and px arbitrarily large. Since M; Y and Z are bounded, Equation (16) gives then ( 1)N @N +1U
@y@zN (Y; Z) 0.
Proof of Proposition 5. The proof is a direct adaptation of the proof of Proposition 4.
Proof of Proposition 7. See Proof of Proposition 9.
Proof of Proposition 9. In an expected utility framework, Nth-degree risk aversion in the direction of y; z is equivalent to
1 2E U y + x2 y; z + x2 z+e2 1 2E U y + x2 y; z + x2 z+e1 > 1 2E U y + x1 y; z + x1 z+e2 1 2E U y + x1 y; z + x1 z+e1 : for all (y; z; x1; x2) 2 R4+ with x2> x1. The previous Inequation is satis…ed for
all x2> x1if and only if E U y + t y; z + t z+e2 E U y + t y; z + t z+e1
is increasing in t or, equivalently, if and only if E f y + t y; z + t z+e2 is
larger than E f y + t y; z + t z+e1 for all (y; z; t) 2 R3+. This is satis…ed
if and only if E [f (y; z +e2)] is larger than E [f (y; z +e1)] for all (y; z) 2 R2+:
Since we want this inequality to be true for all (e1;e2) such that e2 is an
in-crease in Nth-degree risk overe1; this inequality is equivalent, by Lemma 2, to
( 1)Nfz(N ) 0:
Proof of Proposition 12. In an expected utility model, Nth-degree multiplicative-risk attraction in the direction of y; z is equivalent to
1 2E U y + x2 y; z + x2 ze2 1 2E U y + x2 y; z + x2 ze1 > 1 2E U y + x1 y; z + x1 ze2 1 2E U y + x1 y; z + x1 ze1 (17) for all (e1;e2) such that e14N e2 and all (x1; x2) such that 0 < x1 < x2 and
y + xi y 0; i = 1; 2: This inequality is satis…ed for all x2 > x1 if and only if
E U y + t y; z + t ze2 E U y + t y; z + t ze1 is increasing on T (y) =
t 2 R+: y + t y 0 or, equivalently, if and only if E [h (e2; y; z; t)] is larger
than E [h (~1; y; z; t)] for all (y; z) 2 R2+ and for t 2 T (y) where h ( ; y; z; t) = yUy y + t y; z + t z + zUz y + t y; z + t z . By Lemma 2, this last
inequality is satis…ed for every pair (e1;e2) such thate2 is an increase in
Nth-degree risk over e1if and only if ( 1)Nh(N ) 0:
Proof.Simple calculus gives h(N )= tN y Nz @ N +1 U @y@zN+ tN N +1z @ N +1 U @zN +1+N tN 1 Nz @ N U @zN where derivatives of U are taken at y + t y; z + t z . Since t 0 and z 0, our necessary and su¢ cient condition can then be rewritten as ( 1)N t y@N +1U
@y@zN + t z@ N +1 U @zN +1 + N@ N U @zN
0 for all ( ; y; z) 2 R3+and for t 2 T (y): Denoting y+t yby Y and z + t zby Z;
our necessary and su¢ cient condition is equivalent to ( 1)N (Y y)@@y@zN +1NU + (Z z)
@N +1U
@zN +1 + N
@NU
@zN 0 for all (Y; Z) 2 R2
+ and all (y; z) such that y Y and 0 z Z and where
the derivatives are taken at (Y; Z). Taking successively y = Y and z = Z; y = Y and z = 0 we obtain that ( 1)N @@zNNU 0 and ( 1)
N
Z@@zN +1N +1U + N@ NU
@zN 0: Letting y go to 1, we obtain ( 1)N @N +1U
@y@zN 0: Conversely, if these 3 conditions are satis…ed and for a given y Y; we have ( 1)N(Y y)@@y@zN +1NU 0 and
( 1)N (Z z)@ N +1U @zN +1 + N @NU @zN = Z z Z ( 1) N Z@ N +1U @zN +1 + N @NU @zN + z Z( 1) N N @ NU @zN 0
which gives that our necessary and su¢ cient condition is satis…ed.
A far as Nth-degree multiplicative-risk aversion is concerned, it is character-ized by the fact that E U y + t y; z + t ze2 E U y + t y; z + t ze1 is
decreasing on T (y) = t 2 R+: y + t y 0 or, equivalently, by the fact that
E [ h (e2; y; z; t)] is larger than E [ h (~1; y; z; t)] for all (y; z) 2 R2+ and for
t 2 T (y) or, …nally, by ( 1)N +1h(N ) 0: The rest of the proof is then directly adapted from the multiplicative-risk attraction setting.
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